ﻻ يوجد ملخص باللغة العربية
We obtain lower bounds for the maximum dimension of a simple FG-module, where G is a finite group and F is an algebraically closed field of characteristic p. The bounds are described in terms of properties of p-subgroups of G. When p is 2 or p is a Mersenne prime, the bounds take a different form, due to exceptions which arise for such primes.
We give a short proof of the fact that if all characteristic p simple modules of the finite group G have dimension less than p, then G has a normal Sylow p-subgroup.
In this paper we measure how efficiently a finite simple group $G$ is generated by its elements of order $p$, where $p$ is a fixed prime. This measure, known as the $p$-width of $G$, is the minimal $kin mathbb{N}$ such that any $gin G$ can be written
For each $n$ we construct examples of finitely presented $C(1/6)$ small cancellation groups that do not act properly on any $n$-dimensional CAT(0) cube complex.
In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces $mathbb{C}$
Fix an arbitrary finite group $A$ of order $a$, and let $X(n,q)$ denote the set of homomorphisms from $A$ to the finite general linear group ${rm GL}_n(q)$. The size of $X(n,q)$ is a polynomial in $q$. In this note it is shown that generically this p